Optimal computation of brightness integrals parametrized on the unit sphere

Abstract

We compare various approaches to find the most efficient method for the practical computation of the lightcurves (integrated brightnesses) of irregularly shaped bodies such as asteroids at arbitrary viewing and illumination geometries. For convex models, this reduces to the problem of the numerical computation of an integral over a simply defined part of the unit sphere. We introduce a fast method, based on Lebedev quadratures, which is optimal for both lightcurve simulation and inversion in the sense that it is the simplest and fastest widely applicable procedure for accuracy levels corresponding to typical data noise. The method requires no tessellation of the surface into a polyhedral approximation. At the accuracy level of 0.01 mag, it is up to an order of magnitude faster than polyhedral sums that are usually applied to this problem, and even faster at higher accuracies. This approach can also be used in other similar cases that can be modelled on the unit sphere. The method is easily implemented in lightcurve inversion by a simple alteration of the standard algorithm/software.

Compact YORP formulation and stability analysis

“…As pointed out in BRV11, the torque τ for any shape, computed on a grid of (θ, ϕ) by ray-tracing, can, of course, be expressed as a Laplace series by using, e.g., Lebedev-Laikov quadratures (Kaasalainen et al 2012). The coefficients t lm can be used in the following analytical motion-averaged formulae that no longer require convexity, although the relevant t lm tend to be unstable, as discussed in Sect.…”

“…If G(η) and x(η) are defined with spherical harmonics series, τ or T lm can be computed analytically as mentioned earlier, or with quadratures, especially Lebedev-Laikov on S 2 (Kaasalainen et al 2012). The change in T due to shifting the origin to some x 0 ; i.e., x → x − x 0 , is easily computed by writing T → T − ∆T with…”

A fast ellipsoid model for asteroids inverted from lightcurves

“…Lebedev & Laikov (1999) presented a fast method to calculate the surface integral on the unit sphere S by tiling the triangular facets not equally. Kaasalainen et al (2012) applied this technique into their method and confirmed the lebedev quadrature is efficient in the surface integration. The different distributions of triangular facets of the traditional triangularization and lebedev quadrature are shown in Due to that the lebedev quadrature is based on the unit sphere, in our method a curvature function from the surface of ellipsoid E to the unit sphere S is built with the format…”

Cellinoid shape model for multiple light curves

“…Ultimately, following the previous formulas, the total brightness BðJD t ; E 0 ; EÞ in Eq. (6) can be calculated by the discretization of its surface in either of triangularization and Lebedev quadrature (Kaasalainen et al, 2012).…”

“…Furthermore, Kaasalainen et al presented an effective method to reconstruct the shape model by an arbitrary convex surface from many light curves observed in various geometries (Kaasalainen and Lamberg, 1992a,b). Moreover they also numerically improved the performance of the algorithm by employing the efficient Lebedev quadrature (Kaasalainen et al, 2012). Based on Kaasalainen's method, Ďurech built a database to present the shape models and other physical parameters for less than 400 asteroids (Ďurech et al, 2010).…”